Height one specializations of Selmer groups

Palvannan, Bharathwaj

Annales de l'Institut Fourier, Tome 69 (2019) no. 1, pp. 303-334.

Nous donnons des applications de l’étude du comportement des groupes de Selmer sous la spécialisation. Nous considérons les groupes de Selmer associés à de représentations galoisiennes de dimension $4$ provenant

• (i) du produit tensoriel de deux familles cuspidales de Hida $F$ et $G$,
• (ii) de la déformation cyclotomique du dernier,
• (iii) du produit tensoriel d’une forme cuspidale $f$ par une famille de Hida $G$, où $f$ est une spécialisation classique de $F$ de poids $k\ge 2$.

Nous démontrons des théorèmes de contrôle qui relient

• (a) le groupe de Selmer associé au produit tensoriel des familles de Hida $F$ et $G$ au groupe de Selmer associé à sa déformation cyclotomique,
• (b) le groupe de Selmer associé au produit tensoriel de $f$ par $G$ au groupe de Selmer associé au produit tensoriel de $F$ et $G$.

Du côté analytique des conjectures principales, Hida a construit des fonctions $L$ $p$-adiques de Rankin–Selberg à une variable, à deux variables et à trois variables. Nos résultats sur la spécialisation nous permettent de vérifier les résultats de Hida qui relient

• (a) la fonction $L$ $p$-adique à deux variables à la fonction $L$ $p$-adique à trois variables, et
• (b) la fonction $L$ $p$-adique à une variables à la fonction $L$ $p$-adique à deux variables,

et nos théorèmes de contrôle pour les groupes de Selmer sont complètement compatibles avec les conjectures principales.

We provide applications to studying the behavior of Selmer groups under specialization. We consider Selmer groups associated to $4$-dimensional Galois representations coming from

• (i) the tensor product of two cuspidal Hida families $F$ and $G$,
• (ii) its cyclotomic deformation,
• (iii) the tensor product of a cusp form $f$ and the Hida family $G$, where $f$ is a classical specialization of $F$ with weight $k\ge 2$.

We prove control theorems to relate

• (a) the Selmer group associated to the tensor product of Hida families $F$ and $G$ to the Selmer group associated to its cyclotomic deformation, and
• (b) the Selmer group associated to the tensor product of $f$ and $G$ to the Selmer group associated to the tensor product of $F$ and $G$.

On the analytic side of the main conjectures, Hida has constructed one variable, two variable and three variable Rankin–Selberg $p$-adic $L$-functions. Our specialization results enable us to verify that Hida’s results relating

• (a) the two variable $p$-adic $L$-function to the three variable $p$-adic $L$-function, and
• (b) the one variable $p$-adic $L$-function to the two variable $p$-adic $L$-function,

and our control theorems for Selmer groups are completely consistent with the main conjectures.

Classification : 11R23, 11F33, 11F80
Keywords: Iwasawa theory, Hida theory, Selmer groups
Mot clés : théorie d’Iwasawa, théorie de Hida, groupes de Selmer

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